General 1Given five points (P1, P2, P3, P4, P5) in the plane, no three of them collinear, construct the conic section.  What follows is the same 5-point conic that was described in the Newton section. Here another walk-through of Newton’s construction. Construct the quadrilateral P1P2P3P4 using lines, not line segments. Points P1 and P3 will be the poles. Through P5 construct line j14 parallel to P1P4, and line j34 parallel to P3P4. Let j14 meet line P1P2 at A, and let j34 meet line P3P2 at B. Construct line AB.  Construct a circle through P3, and let it be the path of point C. Let P3C intersect j34 at D. Construct DE parallel to AB, meeting j14 at E. Let P1E and P3C meet at F. Construct the locus of F as point C travels its path. This is the conic section, a hyperbola in this case.  For construction of the related objects it will be necessary to have tangent lines through three of the points on curve. Begin with the poles, P1 and P3. Construct line P1P3, which meet meets j34 at G and j14 at H. Through G and H construct lines GJ and HK parallel to AB, where J is on j14 and K is on j34. Lines P1J and P3K are tangent to the curve.  One more tangent line is still needed. For that, let P2 and P4 be the new poles. Through P5, construct line j23 parallel to P2P3. Line j34 has already been constructed, and will be used again. Line j23 meets P1P2 at L, and line j34 meets P1P4 at M. Construct line LM and P2P4. Let line P2P4 meet j23 at R. Construct RS parallel to LM, meeting j34 at S. Line P4S is tangent to the curve at P4.  We now have the conic section and tangents through three of the given points. In this hyperbola the related objects include the center, two axial vertices, two axes, two foci, two directrices, and two asymptotes. They can be constructed using the methods shown in Conic Parameters. If the given points are dragged into positions to result in an ellipse, the related objects will be much different, with four axial vertices and no asymptotes. An ideal construction will result in the correct objects in either case. Drag the given points in the websketch below, and confirm that the construction holds up for every positioning of the given points. |